The long wavelength approximation

The long wavelength approximation is not as much a "mathematical trick" as it is a fundamental part of nature itself. The meaning of that is a the liquid in a pipe of some length compared to the diameter cannot only be treated as a 1D phenomenon, it essentially IS a 1D physical entity. Treating it in full 3D certainly can be done, but the 1D nature will completely overshadow any added wanted increase in accuracy from the 3D approach. The only valuable output from a 3D approach, at least for engineering purposes, will be the 3D approach showing it is essentially a 1D phenomenon.

This has all to do with inertia and the characteristic length and diameter of the pipe. A constraint is also that any externally forced frequency must be lower than first radial resonance frequency, which normally starts in the kHz range.

Why this is so is explained next. We look at equation 3.6 and 3.7 from part 1.

Equation 3.5 is not needed, because the constraint is frequencies below the first radial filled mode. In other words, what we are looking for is when we can use that constrain, and which parameters decides it. This is done using non-dimensional analysis, so we can see the relative magnitude of radial modes vs axial modes. The following non-dimensional variables are used:

These are inserted into B.1 and B.2 and we get:

These two equations are very similar. In fact, they are exactly equal, both mathematically and in magnitude (physically) when:

This means that if we have a pipe, and the L/D ratio is 1/2 (it's half as long as it's diameter), then radial and axial inertia will have equal effect on the dynamics of the system. Is a pipe where the length is only half as long as the diameter really a pipe? Hardly. So we introduce the constraint that L/D >> 1, or conversely that D/L << 1. We then see that the first term in Equation B.6 becomes very small (much less than 1), but nothing happens to the other terms in B.5 or B.6. The larger the ratio L/D becomes, the smaller the inertia term in B.6 becomes.

With L/D = 5, the effect or the radial inertia is already an order of magnitude less than the effect of axial inertia. Another way to see this is to derivate Equation B.5 with respect to r, and B.6 with respect to x, and combine them to get:

This equations shows the magnitude of the change of velocity in radial direction with respect to change in x and the magnitude of change of velocity in axial direction with respect to change in r. When inserting L/D  >> 1 (D/L << 1), the first term simply vanishes. Equation B.7 then becomes:

This equations say that the change of axial velocity with respect to r is zero, thus the axial velocity is constant across the cross section.

A key element of this analysis is of course non-dimensionality, which is a very powerful, and often necessary tool in all of engineering. Non-dimensionality enables us to extract the physical essence that we otherwise would not be able to see.

One question remains. What exactly is L/D in terms of a numerical value so that L/D >> 1? Is it 5? 100? 8.67? There is no simple answer. For instance, a complex piping system may have a few very long pipes, a few rather short ones and lots of pipes with lengths in between. We will then be more interested in the characteristics of the total system, than in each individual pipe. For the vast majority of systems, the longest pipes will decide the total characteristics. If the average L/D for all pipes is larger than say 10, then the 1D phenomenon will be the important one. If the average L/D is smaller than say 2, then this is hardly a piping system anymore, but a complex 3D shape and should be analyzed as such. The boundary conditions are also very important in this respect, as well as the reason for doing a transient analysis in the first place.

This is the case, not only for transient analysis, but also for steady state analysis. After a sharp 10 degree bend, the flow is not considered to be fully reorganized before after 10 diameters in length and so on. This doesn't mean that 1D analysis cannot be done. In most cases it only means that we have to be aware that the accuracy may not be as good as it normally would be. Another question is why would you do a transient analysis in the first place? As long as it is conservative and roughly accurate, it would be difficult to justify the added cost (and added other uncertainties) of a full 3D analysis, unless other considerations, typically FSI, also becomes part of the picture, and failure of the system is unacceptable.